Investigation of Halo Parameter Scaling Relations

Fig. 6.13: Velocity dispersionσ as function of the absolute luminosity. We determine the velocity dis-persion by fitting an SIS out to a scale of 100 h⁻¹kpc to the excess surface mass density∆Σin separate luminosity bins (see Fig. 6.11). We show the combined morphology samples in different environments, magenta triangles (dashed fit-line) in dense environments, green squares (dotted fit-line) in environments of low density and black circles (solid fit-line) in all environments. The results for bright galaxies are in perfect agreement, but for faint, i.e., low mass galaxies we see a mass excess increasing with environment density as already observed before (see Fig. 6.9).

e.g., Saglia et al. 2010 or Bernardi et al. 2010), blue galaxies evolve more rapidly. As for an accurate estimate a star formation history is needed, which cannot be extracted sufficiently well from u^∗g^′r^′i^′z^′ photometry, we use also the evolution L∝(1+z)for blue galaxies as a lower limit. As we see in the right panel of Fig. 6.14, the amplitudes of the scaling increase to values ofσ^∗=150±2 km s⁻¹for the combined lens sample,σ_red^∗ =173±2 km s⁻¹for the red lens sample andσ_blue^∗ =123±3 km s⁻¹ for the blue galaxies. However the slope of the L_r′−σ-relation remains hardly changed. We find for both SED types a scaling behavior corresponding to the Faber-Jackson or Tully-Fisher relation, σred∝L^0.25±0.03 for red andσblue∝L^0.24±0.03 for blue galaxies, while the galaxy type transition for the combined lens sample leads to the steeper scaling relation of σ ∝L^0.29±0.02. The values for the scaling relations also are shown in Table 6.2. A closer inspection of red and blue data points in both panels of Fig. 6.14 reveals that for a luminosity of L∼6−7×10¹⁰h⁻² L_⊙ two red data points are decreased relative to the red SED linear fit and that one blue data point is increased relative to the blue SED linear fit. This could point to a problem in contamination of the red and blue samples with blue and red galaxies at this luminosity.

Assuming an SIS profile (see equation 3.43), the velocity dispersion σhalo obtained from the weak lensing analysis (out to 100 h⁻¹kpc) describes the circular velocity vcirc,halo=σhalo·√

2 of the dark matter halo. The impact of baryons that might change the profile is negligible for the value of σhalosince baryonic effects happen on smaller scales only.

Gerhard et al. (2001) (see their fig. 2) studied the circular velocity curves of local ellipticals with

Fig. 6.14: Velocity dispersion σ as function of the absolute luminosity, without (left panel) and with luminosity evolution according to L∝(1+z)(right panel). Theσ values are obtained by SIS-fits out to a scale of 100 h⁻¹kpc to∆Σin separate luminosity bins (see Fig. 6.11). The complete lens sample is shown in black (circles and solid fit-line), the red galaxy sample in red (triangles and dashed fit-line) and the blue galaxy sample in blue (squares and dotted fit-line). The luminosity evolution only changes the amplitude but not the scaling behavior.

stellar dynamics out to a few (≤3) effective radii. They constrained the anisotropy profilesβ(r)(see Binney & Tremaine 1987) of the stellar orbits and obtained that the mean values forβ are typically between 0.2 and 0.4. The detailed dynamical models yield a relation between the central stellar velocity dispersion and the maximal rotation velocity profile of

σstar=0.66 v^dyn_max. (6.4)

The radii where these maximal velocities are reached are of order 0.5 times the effective radii. The rotation velocities for larger radii (>Re) are flat and have values of≈0.9 v^dynmax.

If one sets these ‘asymptotic values’ equal to the halo circular velocity we obtain vcirc,halo=√

2σ_halo^WL=0.9 v^dyn_max=0.9·1/0.66σstar

or

σhalo^WL=0.96σstar. (6.5)

If one sets the maximal circular velocity equal to the halo circular velocity one obtains

σhalo^WL=1.07σstar. (6.6)

In Fig. 6.15 we compare how the measured velocity dispersionσstarof LRGs compare with predictions from our WL-analysis for red galaxies, based on Eisenstein et al. (2001) and Gallazzi et al. (2006), i.e., we add the best-fitting lines for theσ_halo^WL-luminosity relation, rescaled with 1/0.96 (in magenta)

Fig. 6.15: Velocity dispersionσ for red galaxies as a function of absolute magnitude. The red circles and fit-line show our weak lensing result, divided by 1.07, taking into account that velocity dispersions derived by stellar motions are about 7% lower than halo velocity dispersions. We compare our result with a spectroscopic LRG sample based on Eisenstein et al. (2001), only considering LRGs with redshifts of 0.05<z<0.3 and uncertainties inσ of 0.03<dσstar/σstar<0.1 (black circles and contours), and with the results from Gallazzi et al. (2006) (green dashed fit-line).

and 1/1.07 (in red). We have added the relation between theσstarand evolution corrected luminosities of SDSS-LRGs (Eisenstein et al. 2001) obtained from Gallazzi et al. (2006) as green dashed line.

This relation is however obtained from fitting a linear relation of velocity dispersions vs. absolute magnitude to the overall LRG sample. In the lensing analysis we first average the signal within some (small) luminosity bin and then study the signal. To treat the LRG-galaxies in a similar way we have obtained theσstar-values from the SDSS data base and estimated the luminosity evolved redshift zero absolute magnitudes in the r-band (from SED-fits and a luminosity evolution proportional to 1+z) and obtained the mean stellar velocity dispersion within equidistant luminosity intervals. For this we only include galaxies with redshifts between 0.05 and 0.3 and with secure velocity dispersion estimates 0.03<dσstar/σstar<0.1. The results are plotted with filled black circles, whereas the density contours for all considered galaxies are shown in black. We see that 0.96σstar≤σ_halo^WL≤1.07σstarholds at least for luminosities above 10¹⁰ h⁻² L_⊙. Therefore the halo velocity is between the maximal circular velocity found around 0.5 R_eand 90 per cent of this value which equals the velocity of galaxies at a few effective radii. This indicates that at least for galaxies above this luminosity threshold the halo indeed is isothermal on scales out to 100 h⁻¹kpc.

L-r₂₀₀- and L-M₂₀₀-Scaling Based on Fits to∆Σ

We now investigate halo parameters based on the NFW profile, firstly considering the virial radius r₂₀₀. We calculate the values for r₂₀₀ from a one-dimensional fit, assuming the mass-concentration

relation c∝M^−0.084 of Duffy et al. (2008) (see also equation 3.81). Also in this case we only use scales up to a distance of 100 h⁻¹ kpc from the the∆Σ-profile. The result is shown in Fig. 6.16.

We see that apparently a simple power law is no longer able to fit r₂₀₀ over the whole luminosity range, as, regarding data points with L<10¹⁰h⁻²L_⊙, the scaling behavior of r200becomes clearly shallower. Possible explanations are a contamination in the faint luminosity region by neighboring galaxy halos, a change in the scaling of the virial radius (see, e.g., Kormendy & Bender 2012 due to transition between different red galaxy populations, leading to an almost luminosity independent mass) or a modification of the concentration-mass relation in this regime. The circumstance that the velocity dispersionσ does not show this ‘broken’ scaling behavior indicates that contamination by secondary galaxy halos should not be the reason for this observation. Assuming that the mass concentration relation of Duffy et al. (2008) is correct implies that the r200−L relation cannot be described by a single-power law anymore, but instead with double-power laws and a break at around L=10¹⁰ h⁻² L_⊙, i.e., the mean mass-to-light ratio of galaxies within a luminosity interval would indeed be minimal at this break luminosity. This is in agreement with results from abundance matching (AM) techniques and some satellite kinematic results (see fig. 1 Dutton et al. 2010), in particular with the results of More et al. (2011) (see their fig. 5) who also obtained a change of slope for the red galaxies’ M200−L relation at a luminosity of about 10¹⁰ h⁻² L_⊙. However this result in Fig. 6.16 only holds if the concentration is only weakly changing with virial mass. On the other hand, instead, an approximate single-power law r₂₀₀-luminosity relation could be reconciled, requiring the concentration to steeply rise for luminosities smaller than 10¹⁰ h⁻² L_⊙. We will investigate these two alternatives in more detail in Section 6.4.3. Because of the apparently broken r₂₀₀-luminosity scaling relation we measure the power law slope only for galaxies brighter than 10¹⁰h⁻²L_⊙. For the virial radius we obtain power laws of r^red₂₀₀∝L^0.33±0.04for red and r^blue₂₀₀ ∝L^0.36±0.07for blue galaxies without luminosity evolution and of r₂₀₀^red ∝L^0.38±0.04and r^blue₂₀₀ ∝L^0.40±0.08 for luminosities evolving with(1+z). If galaxies are not separated into blue and red SED types we obtain (for the combined sample) r200∝L^0.39±0.03, ignoring luminosity evolution, and r200∝L^0.37±0.04, assuming a (1+z) scaling. As before the steeper scaling is due to the fact that the amplitudes for the r₂₀₀−L scalings are different for red and blue galaxies and the fractional mix of red and blue galaxies changes as a function of absolute luminosity.

We translate the result for r₂₀₀ to the virial velocity v₂₀₀ in Fig. 6.17 using equation (3.67).

The right panel of Fig. 6.17 shows v200 versus luminosity for our blue galaxy sample (blue data points) and the power law fit for L>10¹⁰ h⁻² L_⊙ (blue dotted line). Reyes et al. (2011) have measured v₂₀₀ for SDSS disk (and thus blue SED-type dominated) galaxies as a function of stellar mass. In order to compare their result to ours we translate their stellar mass estimate (back) to luminosity. For local disk galaxies (the Reyes et al. 2011 disk galaxies have redshifts between 0.02 and 0.1) an average mass-to-light ratio of M_∗/L_r=1 M_⊙/L_⊙ appears to be a good description. On one hand this can be seen in fig. 1 of van Uitert et al. (2011) by comparing their blue histograms on the vertical to the horizontal axis showing the luminosity distribution and stellar mass distribution of blue SDSS-galaxies. This is in agreement with Bell et al. (2003), if one takes into account that our local (see Fig. 5.27) galaxies have a (g−r)-color of approximately 0.3−0.4 at the bright end (which are the galaxies in common with Reyes et al. 2011). The same result is obtained from Kauffmann et al. (2003), fig. 14, upper right panel, taking into account that our local blue galaxies are dominated by absolute magnitudes fainter than Mr^′ =−21. For the three luminosity intervals provided by Reyes et al. (2011) their data points (translated to luminosity) agree well with ours (see

Fig. 6.16: r₂₀₀as a function of absolute luminosity. The left panel shows the result without, the right panel with luminosity evolution L∝(1+z). The red triangles and dashed fit-lines denote red galaxies, the blue squares and dotted fit-lines blue galaxies and the black circles and solid fit-lines all galaxies. We see that a single-power law apparently is no longer able to fit the scaling relation. Therefore only data points with L>10¹⁰h⁻²L_⊙are used for the determination of the scaling relation. For the combined lens sample, r₂₀₀ scales with L^0.39±0.04ignoring and L^0.43±0.04including luminosity evolution.

Fig. 6.17, right panel). We have a larger dynamic range and can extend our analysis down to to a few times 10⁹ L_⊙. In an analogous way we have translated the Dutton et al. (2010) model for the v200-stellar mass relation to the v200-luminosity relation, agreeing well with our result, but possibly showing a slightly shallower slope.

Confident that for the considered absolute magnitude and redshift range we can prop-erly translate our absolute luminosities into stellar mass estimates for red galaxies, we use log₁₀(M_∗) =1.093 log₁₀L_r−0.573 (which was used by Dutton et al. 2010 and derived from Gallazzi et al. 2006), inserting luminosity evolution corrected luminosities. Our results for v200

are shown in red in Fig. 6.17, together with the model of Dutton et al. (2010), being the same to a remarkable level. Only the results for the second and third brightest luminosity interval lie below for reasons we already speculated about. On top we have added the result for vopt as obtained from the Gallazzi et al. (2006)σ−L relation, using the prefactors of Dutton et al. (2010) for the relation between velocity dispersion and rotation velocity. We conclude that for luminosities between 10¹⁰ and 6×10¹⁰h⁻²L_⊙, the mass density profile of ellipticals is not only isothermal out to 100 h⁻¹kpc (as shown before), but also out to the virial radius. For higher luminosities, the virial velocity exceeds the optical velocity.

Finally translating our virial radii into virial masses we show results with and without lumi-nosity evolution correction in the left and right panels of Fig. 6.18. We continue using only galaxies with L>10¹⁰ h⁻² L_⊙ for the power law fits (added as red dashed and blue dotted lines). For the

Fig. 6.17: Circular velocity v₂₀₀as a function of absolute luminosity for our blue galaxy sample (blue filled squares, blue dotted fit-line). Analogously to the fit of r₂₀₀we only use data points with L>10¹⁰h⁻²L_⊙ for the determination of the scaling relation fit. Our measurements agree quite well with the results of Reyes et al. (2011) (green empty squares, green long-dashed line) and Dutton et al. (2010) (magenta dashed-dotted line).

In the left panel we show the circular velocity for our red galaxy sample in red. On top we add the model from Dutton et al. (2010) as a solid line and the result for v_optof Gallazzi et al. (2006) as a dashed line.

combined sample we obtain M₂₀₀ ∝L^1.21±0.10 and M₂₀₀∝L^1.12±0.11 for the case without and with luminosity evolution correction. This scaling agrees with the results of Guzik & Seljak (2002) within their larger uncertainties (M∝L^1.34_r′ ^±0.17). We have further included the results of Hoekstra et al.

(2005) as magenta points, also well agreeing with our blue sample. This agreement appears reason-able since the Hoekstra et al. (2005) sample contains isolated galaxies, thus mostly consisting of blue galaxies. In addition we have considered the excess surface mass density profiles of van Uitert et al.

(2011) (see their fig. 8), and translated them into virial mass estimates in the same way as we did for our work. These estimates are shown as green points. They agree well with our red sample results, again being reasonable since the van Uitert et al. (2011) sample is dominated by red galaxies. All results obtained for r₂₀₀and M₂₀₀are summarized in Table 2.

At last we translate our M200−L relation from the right panel of Fig. 6.18 into the M200 versus stellar mass relation (MSR), again using the relation log₁₀(M_∗) =1.093 log₁₀L_r−0.573 as above.

The result is shown in Fig. 6.19. The virial-to-stellar mass ratio (shown as red points) is almost constant (at ∼100) for a decade in stellar mass (10¹⁰ to 10¹¹ h⁻² M_⊙) and increases for lower stellar masses. This result precisely agrees with the Dutton et al. (2010) model shown as the black solid curve. At the high stellar mass end the MSR appears to only slightly increase (if at all) with stellar mass. This saturation is in agreement with the results of van Uitert et al. (2011) (green points, taken from their fig. 14, and converting their stellar masses to H₀=100 km s⁻¹Mpc⁻¹, as in this Figure the stellar masses are given for H₀=70 km s⁻¹Mpc⁻¹ and the virial masses are given for

Fig. 6.18: M₂₀₀as a function of luminosity. The left panel shows the result without, the right panel with luminosity evolution L∝(1+z). Red triangles and dashed fit-lines denote red galaxies, blue squares and dotted fit-lines blue galaxies and black circles and solid fit-lines all galaxies. We see as expected the same scaling behavior as for r₂₀₀(see Fig. 6.16). Only data points with L>10¹⁰h⁻²L_⊙are used for the determination of the scaling relation. For the complete lens sample the M₂₀₀scales with L^1.21±0.10ignoring and L^1.31±0.13including luminosity evolution. We included the results from van Uitert et al. (2011) in the right panel (green crosses), observing good agreement, given that their analysis describes a red SED-type dominated lens sample.

Fig. 6.19: Stellar Mass versus M₂₀₀/Mstar-ratio for red galaxies converted to z =0. The red trian-gles denote our red galaxies. We have added the results of Mandelbaum et al. (2006c) (open triantrian-gles), Mandelbaum et al. (2008) (open squares) and Dutton et al. (2010) (black solid line), see also fig. 1 in Dutton et al. (2010). We further include the results of van Uitert et al. (2011) from their fig. 14 as green open circles.

Without luminosity evolution, L_r^∗′ =1.6×10¹⁰h⁻²L_r′,⊙

Sample σ^∗[km s⁻¹] ησ r^∗₂₀₀[h⁻¹kpc] ηr₂₀₀ M^∗₂₀₀[10¹¹h⁻¹M_⊙] ηM₂₀₀

All 135±2 0.29±0.02 146±2 0.39±0.03 11.1±0.4 1.21±0.10 Red 162±2 0.24±0.03 177±3 0.33±0.04 18.6±0.8 1.05±0.12 Blue 115±3 0.23±0.03 120±2 0.36±0.07 5.8±0.5 1.14±0.20

With luminosity evolution, L^∗_r′ =1.6×10¹⁰h⁻²L_r′,⊙

Sample σ^∗[km s⁻¹] ησ r^∗₂₀₀[h⁻¹kpc] ηr₂₀₀ M^∗₂₀₀[10¹¹h⁻¹M_⊙] ηM₂₀₀

All 150±2 0.29±0.02 170±2 0.37±0.04 17.0±0.6 1.12±0.11 Red 173±2 0.25±0.03 198±3 0.38±0.04 26.1±1.1 1.17±0.13 Blue 123±3 0.24±0.03 133±3 0.40±0.08 8.7±0.6 1.37±0.25

Table 6.2: Best fits for the scaling relations of the velocity dispersionσ, assuming an SIS and for the r₂₀₀and M₂₀₀, assuming an NFW profile without and with luminosity evolution. The SIS fits have been extracted from all all luminosity bins, the fits for the NFW profiles only include luminosities brighter than L=10¹⁰h⁻¹L_⊙.

H0=100 km s⁻¹Mpc⁻¹ according to van Uitert, private communication) which also saturates at a value of about 100 to 150. The points of van Uitert et al. (2011) for low stellar masses seem however very low, even being below the early results of Mandelbaum et al. (2006c). However, since the van Uitert et al. (2011) M200 versus luminosity relation derived by us from their ∆Σ results agree well with ours, the difference can only be due to a different relation for the stellar masses (especially considering that van Uitert et al. 2011 aim to add up the total stellar mass, i.e., not only that of the central galaxy but also that of its satellites).